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arXiv:1401.7678 · Physical Review B

Quantum spin liquid with a Majorana Fermi surface on the three-dimensional hyperoctagon lattice

Maria Hermanns, Simon Trebst

Published Jun 2, 2014• 114 citations
arXivHTMLPDF

Abstract

Motivated by the recent synthesis of $\ensuremath{\beta}$-Li${}_{2}$IrO${}_{3}$---a spin-orbit entangled $j=1/2$ Mott insulator with a three-dimensional lattice structure of the Ir${}^{4+}$ ions---we consider generalizations of the Kitaev model believed to capture some of the microscopic interactions between the iridium moments on various trivalent lattice structures in three spatial dimensions. Of particular interest is the so-called hyperoctagon lattice---the premedial lattice of the hyperkagome lattice, for which the ground state is a gapless quantum spin liquid where the gapless Majorana modes form an extended two-dimensional Majorana Fermi surface. We demonstrate that this Majorana Fermi surface is inherently protected by lattice symmetries and discuss possible instabilities. We thus provide the first example of an analytically tractable microscopic model of interacting SU(2) spin-$\frac{1}{2}$ degrees of freedom in three spatial dimensions that harbors a spin liquid with a two-dimensional spinon Fermi surface.

DOI
10.1103/physrevb.89.235102

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Motivated by the recent synthesis of \ensuremathβ\ensuremath{\beta}\ensuremathβ-Li2{}_{2}2​IrO3{}_{3}3​---a spin-orbit entangled j=1/2j=1/2j=1/2 Mott insulator with a three-dimensional lattice structure of the Ir4+{}^{4+}4+ ions---we consider generalizations of the Kitaev model believed to capture some of the microscopic interactions between the iridium moments on various trivalent lattice structures in three spatial dimensions. Of particular interest is the so-called hyperoctagon lattice---the premedial lattice of the hyperkagome lattice, for which the ground state is a gapless quantum spin liquid where the gapless Majorana modes form an extended two-dimensional Majorana Fermi surface. We demonstrate that this Majorana Fermi surface is inherently protected by lattice symmetries and discuss possible instabilities. We thus provide the first example of an analytically tractable microscopic model of interacting SU(2) spin-12\frac{1}{2}21​ degrees of freedom in three spatial dimensions that harbors a spin liquid with a two-dimensional spinon Fermi surface.
Three-dimensional (3D) variants of the Kitaev model can harbor gapless spin liquids with a Majorana Fermi surface on certain tricoordinated lattice structures such as the recently introduced hyperoctagon lattice. Here, we investigate Fermi surface instabilities arising from additional spin exchange terms (such as a Heisenberg coupling) which introduce interactions between the emergent Majorana fermion degrees of freedom. We show that independent of the sign and structure of the interactions, the Majorana surface is always unstable. Generically, the system spontaneously doubles its unit cell at exponentially small temperatures and forms a spin liquid with line nodes. Depending on the microscopics, further symmetries of the system can be broken at this transition. These spin-Peierls instabilities of a 3D spin liquid are closely related to BCS instabilities of fermions.
The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.
In quantum magnetism, Kitaev's honeycomb model is well known for its gapped and gapless spin liquid ground states in which the elementary spin degrees of freedom fractionalize into Majorana fermions and a Z2{\mathbb{Z}}_{2}Z2​ gauge field. In this paper, the authors discuss the gapless spin liquids arising in three-dimensional Kitaev models and show that the gapless Majorana fermions form metallic states which, depending on the underlying lattice structure, exhibit Fermi surfaces, nodal lines, or Weyl points.
Recent experiments indicate that Na4Ir3O8, a material in which s=1/2 Ir local moments live on a three-dimensional "hyperkagome" lattice of corner-sharing triangles, may have a quantum spin liquid ground state with gapless spin excitations. Using a combination of exact diagonalization, symmetry analysis of fermionic mean field ground states and Gutzwiller projected variational wave functions studies, we propose a quantum spin liquid with spinon Fermi surfaces as a favorable candidate for the ground state of the Heisenberg model on this lattice. We point out implications of this proposal for thermodynamic properties and discuss possible weak instabilities of the spinon Fermi surfaces.
The fractionalization of quantum numbers in interacting quantum many-body systems is a central motif in condensed-matter physics with prominent examples including the fractionalization of the electron in quantum Hall liquids or the emergence of magnetic monopoles in spin-ice materials. Here, we discuss the fractionalization of magnetic moments in three-dimensional Kitaev models into Majorana fermions (and a Z_{2} gauge field) and their emergent collective behavior. We analytically demonstrate that the Majorana fermions form a Weyl superconductor for the Kitaev model on the recently synthesized hyperhoneycomb structure of β-Li_{2}IrO_{3} when applying a magnetic field. We characterize the topologically protected bulk and surface features of this state, which we dub a Weyl spin liquid, including thermodynamic and transport signatures.
Motivated by recent experiments on $\ensuremath{\beta}$-Li2{}_{2}2​IrO3{}_{3}3​, we study the phase diagram of the Heisenberg-Kitaev model on a three-dimensional lattice of tricoordinated Ir4+{}^{4+}4+, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir4+{}^{4+}4+ ions carrying Jeff=1/2{J}_{\mathrm{eff}}=1/2Jeff​=1/2 moments, suggests that the Heisenberg-Kitaev model may effectively capture the low-energy spin-physics of the system in the strong-coupling limit. Using a combination of semiclassical analysis, exact solution, and slave-fermion mean-field theory, we find, in addition to the spin liquid, four different magnetically ordered phases depending on the parameter regime. All four magnetic phases---the N\'eel, the polarized ferromagnet, the skew-stripy, and the skew-zig-zag---have collinear spin ordering. The three-dimensional Z2{Z}_{2}Z2​ spin liquid, which extends over an extended parameter regime around the exactly solvable Kitaev point, has a gapless Majorana mode with a deformed Fermi circle (codimensions, dc=2{d}_{c}=2dc​=2). We discuss the effect of the magnetic field and finite temperature on different phases that may be relevant for future experiments.